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Digital SAT · Circles

How to Solve Degrees and Radians on the Digital SAT

Degrees and radians are two scales for the same angle. Hold onto one fact and you rarely need a formula: a straight angle, 180 degrees, is pi radians. Halve it for 90 degrees as pi over 2, take a third for 60 degrees as pi over 3, and so on. If you do want the rule, multiply degrees by pi over 180 to get radians, or radians by 180 over pi to go back.

Written from Perfect1600’s analysis of every degrees and radians question in our bank·Method checked against the current Bluebook test
per test
less than 1 per test
per test
typical difficulty
Medium to hard
typical difficulty
practice questions
70
practice questions
How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

180° = π
One anchor covers it

A straight angle in both units. Every common SAT angle is a fraction of it.

1 step
No calculator needed

A single multiplication by pi over 180 or 180 over pi does the whole job.

47%
Grid-ins

Almost half accept a typed answer, so a flipped factor has no choice to catch it.

How to recognize degrees and radians questions

  • An angle is stated in one unit and the answer is wanted in the other.
  • Radians show up written with pi, like pi over 3, while degrees carry the little circle symbol.
  • A trig or circle problem quietly mixes the two units.
  • The choices are angle measures in whichever unit you are converting to.

Why students miss these

There is really only one moving part, the conversion factor, and it is the thing people flip. Going to radians you want pi over 180, but under time pressure the factor gets inverted to 180 over pi, and the answer comes out wildly off. The other slip is quieter: a radian answer that drops its pi and turns into a bare decimal. Both disappear if you sanity-check against the anchor, since 90 degrees has to land on pi over 2, not some random number.

The step-by-step method

  1. 1

    Notice which way you are going

    From degrees to radians, or radians to degrees. The direction picks the factor.

  2. 2

    Multiply by the right factor

    Degrees to radians is times pi over 180; radians to degrees is times 180 over pi.

  3. 3

    Let the unit cancel

    Write the factor so the unit you are leaving divides out and the target unit is what remains.

  4. 4

    Simplify and keep pi

    Reduce the fraction. A radian result keeps its pi; a degree result comes out as a plain number.

Worked examples

Easy example
A central angle measures 60∘60^\circ. What is its measure in radians?
A
π12\frac{\pi}{12}
π3\frac{\pi}{3}
C
π2\frac{\pi}{2}
D
2π3\frac{2\pi}{3}

A: Incorrect. This is the radian measure of 15∘15^\circ.

B: Correct. 60×π180=π360 \times \frac{\pi}{180} = \frac{\pi}{3}.

C: Incorrect. This is the radian measure of 90∘90^\circ.

D: Incorrect. This is the radian measure of 120∘120^\circ.

Explanation

60∘×π180=π360^\circ \times \frac{\pi}{180} = \frac{\pi}{3}.

Medium example
A right triangle has an acute angle of 30∘30^\circ. What is this angle's measure in radians?
π6\frac{\pi}{6}
B
π4\frac{\pi}{4}
C
π3\frac{\pi}{3}
D
π2\frac{\pi}{2}

A: Correct. 30×π180=π630 \times \frac{\pi}{180} = \frac{\pi}{6}.

B: Incorrect. π4\frac{\pi}{4} corresponds to 45∘45^\circ.

C: Incorrect. π3\frac{\pi}{3} corresponds to 60∘60^\circ.

D: Incorrect. π2\frac{\pi}{2} corresponds to 90∘90^\circ.

Explanation

30×π180=π630 \times \frac{\pi}{180} = \frac{\pi}{6} radians.

Hard example
A central angle measures 13π18\frac{13\pi}{18} radians. What is its measure in degrees?
A
110
B
120
130
D
140

A: Incorrect. This corresponds to 11π18\frac{11\pi}{18}.

B: Incorrect. This corresponds to 2π3\frac{2\pi}{3}.

C: Correct. 13π18×180π=130\frac{13\pi}{18} \times \frac{180}{\pi} = 130.

D: Incorrect. This corresponds to 7π9\frac{7\pi}{9}.

Explanation

13π18×180π=13×18018=130\frac{13\pi}{18} \times \frac{180}{\pi} = \frac{13 \times 180}{18} = 130.

The common traps

PatternWhat it doesThe tell
Factor flippedMultiplied by 180 over pi on the way to radians, or the mirror image.Radians come from pi over 180; degrees come from 180 over pi.
Pi went missingWrote a radian measure as a plain decimal.SAT radian answers are exact and carry a pi, like pi over 6.
Lost the anchorForgot that a straight angle is both 180 degrees and pi radians.Test the result: 90 degrees should be pi over 2, 45 degrees pi over 4.

Try it: two real questions

Question 1easy

What is the radian measure equivalent to 180∘180^\circ?

Question 2easy

Convert 90∘90^\circ to radians.

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Often confused with

Circles
Arc Length and Sector Area
Right-Triangle Trigonometry
Cofunctions and Complementary Angles
Right-Triangle Trigonometry
Trigonometric Ratios (SOH-CAH-TOA)

Related reading

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Which SAT skills move your score most

Common questions

How do you convert degrees to radians?

Multiply by pi over 180. Sixty degrees becomes 60 times pi over 180, which reduces to pi over 3 radians.

How do you convert radians to degrees?

Multiply by 180 over pi. Pi over 4 becomes pi over 4 times 180 over pi, and the pi cancels to leave 45 degrees.

Is there a shortcut for degree-radian conversions?

Anchor on 180 degrees equals pi radians. From there the common SAT angles fall out: 90 is pi over 2, 60 is pi over 3, 45 is pi over 4, 30 is pi over 6.

Should radian answers keep pi on the SAT?

Almost always. Radian measures are written exactly, as pi over 6 rather than 0.52, so leave the pi in your answer unless a decimal is specifically requested.

Practice degrees and radians the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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